Y. Achdou, F. Camilli, A. Cutri, and A. N. Tchou, Hamilton-Jacobi equations constrained on networks, Nonlinear Differential Equations Appl, pp.413-445, 2013.
DOI : 10.1007/s00030-012-0158-1
URL : https://hal.archives-ouvertes.fr/hal-00656919/document

J. Aubin and A. A. Cellina, Differential inclusions, Comprehensive Stud, Math, vol.264, 1984.

G. Barles, A. Briani, and A. E. Chasseigne, A Bellman approach for two-domains optimal control problems in R N , ESAIM Control Optim, Calc. Var, vol.19, pp.710-739, 2013.

G. Barles, A. Briani, and A. E. Chasseigne, A Bellman Approach for Regional Optimal Control Problems in $\mathbb{R}^N$, SIAM Journal on Control and Optimization, vol.52, issue.3, pp.1712-1744, 2014.
DOI : 10.1137/130922288

G. Barles and A. E. Chasseigne, Almost) everything you always wanted to know about deterministic control problems in stratified domains, Networks and heterogenous media 10, pp.809-836, 2015.

M. Bardi and A. , Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton- Jacobi-Bellman Equations, Systems Control Founda. and Appl, 1997.

A. Briani and A. A. Davini, Monge solutions for discontinuous Hamiltonians, ESAIM: Control, Optimisation and Calculus of Variations, vol.11, issue.2, pp.229-251, 2005.
DOI : 10.1051/cocv:2005004
URL : https://hal.archives-ouvertes.fr/hal-00977661

A. Bressan and A. Y. Hong, Optimal control problems for control systems on stratified domains, Network and Heterogeneous Media, pp.313-331, 2007.

R. C. Barnard and A. P. Wolenski, Flow invariance on stratified domains, Set-Valued Variational Anal, pp.377-403, 2013.
DOI : 10.1007/s11228-013-0230-y
URL : http://arxiv.org/abs/1208.4742

F. Camilli and A. C. Marchi, A comparison among various notions of viscosity solution for Hamilton???Jacobi equations on networks, Journal of Mathematical Analysis and Applications, vol.407, issue.1, pp.112-118, 2013.
DOI : 10.1016/j.jmaa.2013.05.015

F. Camilli and A. A. Siconolfi, Hamilton-Jacobi equations with measurable dependence on the state variable Advances in Differential Equations, pp.733-768, 2003.

F. H. Clarke, Optimization and Nonsmooth Analysis, 1990.
DOI : 10.1137/1.9781611971309

F. H. Clarke, Functional analysis, calculus of variations and optimal control, Graduate Text in Mathematics, vol.264, 2013.
DOI : 10.1007/978-1-4471-4820-3
URL : https://hal.archives-ouvertes.fr/hal-00865914

F. H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial Mathematics, 1990.
DOI : 10.1137/1.9781611971309

F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and A. P. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol.178
DOI : 10.1007/978-0-387-30440-3_370
URL : https://hal.archives-ouvertes.fr/hal-00863298

G. Dal-maso and A. H. Frankowska, Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities, ESAIM: Control, Optimisation and Calculus of Variations, vol.5, pp.369-394, 2000.
DOI : 10.1051/cocv:2000114

A. F. Filippov, Differential equations with discontinuous right-hand side, 1988.
DOI : 10.1090/trans2/042/13

H. Frankowska, Lower Semicontinuous Solutions of Hamilton???Jacobi???Bellman Equations, SIAM Journal on Control and Optimization, vol.31, issue.1, pp.257-272, 1993.
DOI : 10.1137/0331016

H. Frankowska and S. Plaskacz, Semicontinuous Solutions of Hamilton???Jacobi???Bellman Equations with Degenerate State Constraints, Journal of Mathematical Analysis and Applications, vol.251, issue.2, pp.818-838, 2000.
DOI : 10.1006/jmaa.2000.7070

C. Hermosilla and A. H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, Journal of Differential Equations, vol.258, issue.4, pp.1430-1460, 2015.
DOI : 10.1016/j.jde.2014.11.001
URL : https://hal.archives-ouvertes.fr/hal-00955921

C. Hermosilla, H. Zidani, and A. P. Wolenski, The mayer and minimum time problems with stratified state constraints, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01517510

H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations, Ann. Sc. Norm. Sup. Pisa (IV), vol.16, pp.105-135, 1989.

C. Imbert and A. R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, -Jacobi equations on networks
URL : https://hal.archives-ouvertes.fr/hal-00832545

C. Imbert, R. Monneau, A. H. Zidani, and . Hamilton, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM: Control, Optimisation and Calculus of Variations, vol.19, issue.1, pp.129-166, 2013.
DOI : 10.1051/cocv/2012002
URL : https://hal.archives-ouvertes.fr/hal-00569010

Z. Rao, A. Siconolfi, and A. H. Zidani, Transmission conditions on interfaces for Hamilton???Jacobi???Bellman equations, Journal of Differential Equations, vol.257, issue.11, pp.3978-4014, 2014.
DOI : 10.1016/j.jde.2014.07.015
URL : https://hal.archives-ouvertes.fr/hal-00820273

Z. Rao and A. H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, Control and Optimization with PDE Constraints, Internat. Ser. Numer. Math, vol.164, pp.93-116, 2013.
DOI : 10.1007/978-3-0348-0631-2_6
URL : https://hal.archives-ouvertes.fr/hal-00803108/file/Rao-Zidani-FinalVersion.pdf

P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian, Indiana University Mathematics Journal, vol.51, issue.2, pp.451-477, 2002.
DOI : 10.1512/iumj.2002.51.2105
URL : http://doi.org/10.1512/iumj.2002.51.2105

P. Wolenski and A. Y. Zhuang, Proximal Analysis and the Minimal Time Function, SIAM Journal on Control and Optimization, vol.36, issue.3, pp.1048-1072, 1998.
DOI : 10.1137/S0363012996299338
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.117.5607